A semi-analytical solution method for two dimensional Helmholtz Equation

Abstract A semi-analytical solution method, the so-called Scaled Boundary Finite Element Method (SBFEM), is developed for the two-dimensional Helmholtz equation. The new method is applicable to two-dimensional computational domains of any shape including unbounded domains. The accuracy and efficiency of this method are illustrated by numerical examples of wave diffraction around vertical cylinders and harbour oscillation problems. The computational results are compared with those obtained using analytical methods, numerical methods and physical experiments. It is found that the present method is completely free from the irregular frequency difficulty that the conventional Green’s Function Method (GFM) often encounters. It is also found that the present method does not suffer from computational stability problems at sharp corners, is able to resolve velocity singularities analytically at such corners by choosing the structure surfaces as side-faces, and produces more accurate solutions than conventional numerical methods with far less number of degrees of freedom. With these attractive attributes, the scaled boundary finite element method is an excellent alternative to conventional numerical methods for solving the two-dimensional Helmholtz equation.

[1]  H. S. Chen,et al.  OSCILLATIONS AND WAVE FORCES IN A MAN-MADE HARBOR IN THE OPEN SEA , 1976 .

[2]  J. Lee,et al.  Wave-induced oscillations in harbours of arbitrary geometry , 1971, Journal of Fluid Mechanics.

[3]  Chongmin Song,et al.  The scaled boundary finite-element method—alias consistent infinitesimal finite-element cell method—for elastodynamics , 1997 .

[4]  B. Teng,et al.  A composite numerical model for wave diffraction in a harbor with varying water depth , 2004 .

[5]  P. Bettess,et al.  A new mapped infinite wave element for general wave diffraction problems and its validation on the ellipse diffraction problem , 1998 .

[6]  M. Rahman,et al.  Wave diffraction due to large offshore structures: A boundary element analysis , 1992 .

[7]  Minoo H. Patel.  Dynamics of Offshore Structures , 1989 .

[8]  Joseph E. Murphy,et al.  Integral Equation Failure in Wave Calculations , 1978 .

[9]  John P. Wolf,et al.  Dynamic‐stiffness matrix of unbounded soil by finite‐element multi‐cell cloning , 1994 .

[10]  Y. Sibuya.  Second order linear ordinary differential equations containing a large parameter , 1958 .

[11]  Bin Teng,et al.  A modified scaled boundary finite-element method for problems with parallel side-faces. Part II. Application and evaluation. , 2005 .

[12]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[13]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.

[14]  Carlos Alberto Brebbia,et al.  Diffraction of water waves for vertical cylinders using boundary elements , 1983 .

[15]  John P. Wolf,et al.  The scaled boundary finite-element method: analytical solution in frequency domain , 1998 .

[16]  O. C. Zienkiewicz,et al.  Numerical methods in offshore engineering , 1978 .

[17]  R C MacCamy,et al.  Wave forces on piles: a diffraction theory , 1954 .

[18]  Andrew Deeks,et al.  Potential flow around obstacles using the scaled boundary finite‐element method , 2003 .

[19]  Carlos Alberto Brebbia,et al.  Basic principles and applications , 1984 .

[20]  J. Wolf,et al.  Finite-element modelling of unbounded media , 1996 .

[21]  Chih-Kang Lee,et al.  A DIRECT BOUNDARY-ELEMENT METHOD FOR THREE-D WAVE DIFFRACTION AND RADIATION PROBLEMS , 1988 .

[22]  Paul D. Sclavounos,et al.  Removing the irregular frequencies from integral equations in wave-body interactions , 1989, Journal of Fluid Mechanics.

[23]  O. C. Zienkiewicz,et al.  Diffraction and refraction of surface waves using finite and infinite elements , 1977 .

[24]  F. John.  On the motion of floating bodies. I , 1949 .

[25]  J. Wolf,et al.  The scaled boundary finite-element method – a primer: derivations , 2000 .

[26]  Fritz John,et al.  On the motion of floating bodies II. Simple harmonic motions , 1950 .

[27]  Bin Teng,et al.  A modified scaled boundary finite-element method for problems with parallel side-faces. Part I. Theoretical developments , 2005 .

[28]  J. Wolf,et al.  The scaled boundary finite element method , 2004 .

[29]  Per A. Madsen,et al.  An efficient finite-difference approach to the mild-slope equation , 1987 .

[30]  Guo-ping Miao,et al.  Water Wave Interaction of Twin Large Scale Caissons with a Small Gap Between , 2001 .